#6407875

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#36696	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26,21,24,29,32)(19,20,25,28,27,23,30,31);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,20)(21,27)(22,25)(23,24)(26,28)(29,30)(31,32);(3,8)(5,7)(10,14)(11,15)(17,19)(18,23)(20,24)(21,27)(22,25)(26,31)(28,32)(29,30);(2,6)(3,8)(10,14)(13,16)(18,24)(19,27)(25,30)(26,32);(17,20,22,28,21,23,29,31)(18,25,26,27,24,30,32,19)

Polycyclic presentation:

2^9|a^2=h,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=cfgi,d^b=dfi,d^c=di,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6315071);